Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

A quadratic equation is one of the most important topics in IGCSE Mathematics. It appears in algebra, graphs, geometry, and many real-life problems. Understanding how to solve quadratic equations confidently will help you score high in your Edexcel exam.

二次方程是 IGCSE 数学中最重要的内容之一。它出现在代数、图像、几何以及许多实际问题中。自信地掌握解二次方程的方法，将帮助你在 Edexcel 考试中取得高分。

1. What Is a Quadratic Equation? | 什么是二次方程？

A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b and c are numbers, and a ≠ 0. The highest power of x is 2, which is why it is called “quadratic”.

二次方程是形如 ax² + bx + c = 0 的方程，其中 a、b、c 是数字，且 a ≠ 0。未知数 x 的最高次数为 2，因此被称为“二次”。

For example: 2x² + 3x – 5 = 0, x² – 4 = 0, and 3x² = 6x are all quadratic equations.

例如：2x² + 3x – 5 = 0、x² – 4 = 0 和 3x² = 6x 都是二次方程。

Some equations can be rearranged into this standard form before solving. Always make sure one side is zero before using factorisation or the quadratic formula.

有些方程需要先变形为标准形式再求解。在使用因式分解或求根公式前，一定要确保一边为零。

2. Solving by Factorisation | 因式分解法

Factorisation is the quickest method when the quadratic expression can be written as a product of two linear factors. For example, x² + 5x + 6 = 0 can be factorised as (x + 2)(x + 3) = 0.

当二次表达式可以写成两个一次因式的乘积时，因式分解是最快捷的方法。例如，x² + 5x + 6 = 0 可以因式分解为 (x + 2)(x + 3) = 0。

If a product of two factors is zero, then at least one factor must be zero. So x + 2 = 0 or x + 3 = 0, giving x = -2 or x = -3.

如果两个因式的乘积为零，那么至少有一个因式为零。所以 x + 2 = 0 或 x + 3 = 0，得到 x = -2 或 x = -3。

Steps for factorising a quadratic ax² + bx + c:

因式分解二次式 ax² + bx + c 的步骤：

Find two numbers that multiply to give ac and add to give b.
找到两个数，它们的乘积等于 ac，和等于 b。
Split the middle term using these two numbers.
用这两个数拆分中间项。
Factor by grouping.
分组因式分解。
Set each bracket equal to zero.
令每个括号等于零。

For example: 2x² + 7x + 3 = 0. Two numbers: 1 and 6 (1 × 6 = 6 = ac, 1 + 6 = 7 = b). Then 2x² + 1x + 6x + 3 = 0, so x(2x + 1) + 3(2x + 1) = 0, giving (2x + 1)(x + 3) = 0, hence x = -1/2 or x = -3.

例如：2x² + 7x + 3 = 0。两个数：1 和 6（1 × 6 = 6 = ac，1 + 6 = 7 = b）。于是 2x² + 1x + 6x + 3 = 0，即 x(2x + 1) + 3(2x + 1) = 0，得到 (2x + 1)(x + 3) = 0，因此 x = -1/2 或 x = -3。

If the coefficient of x² is 1, you can simply look for two numbers that multiply to c and add to b.

如果 x² 的系数为 1，你只需找两个数，它们相乘等于 c，相加等于 b。

3. The Quadratic Formula | 求根公式

When a quadratic cannot be factorised easily, the quadratic formula gives the solution directly.

当二次方程不易因式分解时，求根公式可以直接给出解。

x = (−b ± √(b² − 4ac)) / (2a)

Here a, b and c come from the equation ax² + bx + c = 0. Substitute the values into the formula and simplify.

其中 a、b、c 来自方程 ax² + bx + c = 0。将数值代入公式并化简。

For example, solve 2x² + 3x − 2 = 0. Here a = 2, b = 3, c = −2.

例如，解 2x² + 3x − 2 = 0。这里 a = 2，b = 3，c = −2。

Substitute: x = (−3 ± √(3² − 4 × 2 × (−2))) / (2 × 2) = (−3 ± √(9 + 16)) / 4 = (−3 ± √25) / 4 = (−3 ± 5) / 4.

代入：x = (−3 ± √(3² − 4 × 2 × (−2))) / (2 × 2) = (−3 ± √(9 + 16)) / 4 = (−3 ± √25) / 4 = (−3 ± 5) / 4。

So x = (−3 + 5)/4 = 1/2, or x = (−3 − 5)/4 = −2. The solutions are x = 1/2 and x = −2.

所以 x = (−3 + 5)/4 = 1/2，或 x = (−3 − 5)/4 = −2。解为 x = 1/2 和 x = −2。

Remember to check if b² − 4ac is negative before using the formula. If it is negative, there are no real solutions.

在使用公式前，记得检查 b² − 4ac 是否为负。如果为负，则没有实数解。

4. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form p(x + q)² + r = 0. This is useful for solving equations and finding the turning point of a quadratic graph.

配方法将二次式改写成 p(x + q)² + r = 0 的形式。这在解方程和求二次图像顶点时非常有用。

Start with ax² + bx + c = 0. If a ≠ 1, divide by a first. Then take half of the coefficient of x, square it, and add and subtract it inside the equation.

从 ax² + bx + c = 0 开始。如果 a ≠ 1，先除以 a。然后取 x 系数的一半，平方，并在等式中加上和减去这个平方。

Example: x² + 6x + 2 = 0. Half of 6 is 3, square is 9. So x² + 6x + 9 − 9 + 2 = 0 ⇒ (x + 3)² − 7 = 0. Then (x + 3)² = 7, so x + 3 = ±√7, hence x = −3 ± √7.

例如：x² + 6x + 2 = 0。6 的一半是 3，平方为 9。所以 x² + 6x + 9 − 9 + 2 = 0 ⇒ (x + 3)² − 7 = 0。于是 (x + 3)² = 7，所以 x + 3 = ±√7，因此 x = −3 ± √7。

When the coefficient of x² is not 1, factor it out first. For example, 2x² + 8x + 5 = 0 becomes 2(x² + 4x) + 5 = 0. Then complete the square inside the bracket: 2[(x + 2)² − 4] + 5 = 0 ⇒ 2(x + 2)² − 8 + 5 = 0 ⇒ 2(x + 2)² − 3 = 0. So (x + 2)² = 3/2, giving x = −2 ± √(3/2).

当 x² 的系数不是 1 时，先提取出来。例如，2x² + 8x + 5 = 0 变成 2(x² + 4x) + 5 = 0。然后在括号内配方：2[(x + 2)² − 4] + 5 = 0 ⇒ 2(x + 2)² − 8 + 5 = 0 ⇒ 2(x + 2)² − 3 = 0。所以 (x + 2)² = 3/2，得到 x = −2 ± √(3/2)。

Completing the square also helps you find the minimum or maximum point of a quadratic curve.

配方法还能帮助你找到二次曲线的最小值或最大值点。

5. The Discriminant | 判别式

In the quadratic formula, the expression inside the square root, b² − 4ac, is called the discriminant. It tells you how many real solutions the equation has.

在求根公式中，根号内的表达式 b² − 4ac 被称为判别式。它告诉我们方程有多少个实数解。

If b² − 4ac > 0, there are two distinct real solutions.
如果 b² − 4ac > 0，方程有两个不同的实数解。
If b² − 4ac = 0, there is exactly one real solution (a repeated root).
如果 b² − 4ac = 0，方程恰好有一个实数解（重根）。
If b² − 4ac < 0, there are no real solutions.
如果 b² − 4ac < 0，方程没有实数解。

For example, x² − 4x + 4 = 0 has discriminant (−4)² − 4 × 1 × 4 = 16 − 16 = 0, so it has one repeated root x = 2.

例如，x² − 4x + 4 = 0 的判别式为 (−4)² − 4 × 1 × 4 = 16 − 16 = 0，所以它有一个重根 x = 2。

The discriminant also tells you how many x-intercepts a quadratic graph has. A positive discriminant means two intercepts with the x-axis, zero means one intercept, and negative means no intercept.

判别式还告诉我们二次图像与 x 轴的交点个数。判别式为正表示有两个交点，为零表示有一个交点，为负表示没有交点。

6. Solving by Drawing Graphs | 图像法

Sometimes you may be asked to solve a quadratic equation by reading the graph of y = ax² + bx + c. The solutions are the x-values where the graph crosses the x-axis.

有时你可能会被要求通过读取 y = ax² + bx + c 的图像来解二次方程。解就是图像与 x 轴交点的 x 值。

To draw a quadratic graph, plot at least five points, including the turning point and points on both sides of it. Then draw a smooth curve through the points.

要画二次图像，至少绘制五个点，包括顶点和两侧的点。然后用平滑曲线穿过这些点。

For example, the graph of y = x² − 2x − 3 crosses the x-axis at x = −1 and x = 3. Those are the solutions to x² − 2x − 3 = 0.

例如，y = x² − 2x − 3 的图像在 x = −1 和 x = 3 处穿过 x 轴。这就是方程 x² − 2x − 3 = 0 的解。

If the equation is not in the form y = ax² + bx + c, you may need to rearrange it or solve a pair of equations graphically.

如果方程不是 y = ax² + bx + c 的形式，你可能需要重新整理，或通过图像解一对方程组。

7. Quadratic Graphs and the Turning Point | 二次图像与顶点

The graph of a quadratic function is a parabola. It is symmetric about a vertical line, and the highest or lowest point is called the turning point.

二次函数的图像是抛物线。它关于一条竖直线对称，最高点或最低点称为顶点。

If the coefficient of x² is positive, the parabola opens upwards and the turning point is a minimum. If the coefficient is negative, it opens downwards and the turning point is a maximum.

如果 x² 的系数为正，抛物线开口向上，顶点是最小值。如果系数为负，抛物线开口向下，顶点是最大值。

The x-coordinate of the turning point is x = −b / (2a). To find the y-coordinate, substitute this x-value into the equation.

顶点的 x 坐标为 x = −b / (2a)。要得到 y 坐标，将此 x 值代入原方程。

For example, y = x² − 4x + 5. Here a = 1, b = −4, so x = 4 / (2 × 1) = 2. Then y = 2² − 4 × 2 + 5 = 1. So the turning point is (2, 1).

例如，y = x² − 4x + 5。这里 a = 1，b = −4，所以 x = 4 / (2 × 1) = 2。然后 y = 2² − 4 × 2 + 5 = 1。因此顶点为 (2, 1)。

Using completing the square, y = (x − 2)² + 1, which directly shows the turning point (2, 1).

利用配方法，y = (x − 2)² + 1，直接显示出顶点 (2, 1)。

8. Word Problems with Quadratics | 二次方程应用题

Many real-world problems produce quadratic equations. You must first translate the problem into an equation, then solve it, and finally check that your answers make sense.

许多现实问题会产生二次方程。你必须先将问题转化为方程，然后求解，最后检查答案是否合理。

Common types include area problems, projectile motion, and number puzzles. For area, length × width = area; for motion, height = initial height + velocity × time − 5t² (on Earth).

常见类型包括面积问题、抛体运动和数字谜题。对面积，长 × 宽 = 面积；对运动，高度 = 初始高度 + 速度 × 时间 − 5t²（在地球上）。

Example: A rectangle has length 2 cm more than its width. Its area is 24 cm². Let width = x, then length = x + 2. So x(x + 2) = 24 ⇒ x² + 2x − 24 = 0 ⇒ (x + 6)(x − 4) = 0 ⇒ x = 4 or x = −6. Since width cannot be negative, width = 4 cm and length = 6 cm.

例：一个长方形的长比宽多 2 cm，面积为 24 cm²。设宽为 x，则长为 x + 2。所以 x(x + 2) = 24 ⇒ x² + 2x − 24 = 0 ⇒ (x + 6)(x − 4) = 0 ⇒ x = 4 或 x = −6。因为宽不能为负，所以宽 = 4 cm，长 = 6 cm。

Always reject negative or impossible answers in word problems, even if they are mathematically valid.

在应用题中，即使负根或不合理的根在数学上成立，也要舍弃。

9. Common Mistakes to Avoid | 常见错误提醒

Students often lose marks in quadratic questions because of small errors. Here are some common mistakes and how to avoid them.

学生在二次方程题目中常因小错误而失分。以下是一些常见错误及避免方法。

Forgetting to write the equation in the form ax² + bx + c = 0 before factorising.
在因式分解前忘记把方程写成 ax² + bx + c = 0 的形式。
Using the quadratic formula incorrectly because the signs of b or c are wrong.
因为 b 或 c 的符号错误而错用求根公式。
Dropping the ± sign when taking square roots.
在开平方时忽略 ± 号。
Thinking that (a + b)² = a² + b². This is false.
错误认为 (a + b)² = a² + b²。这是错误的。
Not checking solutions by substituting back into the original equation.
不将解代回原方程进行检验。

To avoid these mistakes, write carefully, show every step, and practice regularly.

为了避免这些错误，请书写工整，展示每一步，并经常练习。

10. Exam Tips for Edexcel | Edexcel 考试建议

In the Edexcel IGCSE exam, quadratic questions appear in both Paper 1 and Paper 2, and may be worth 2 to 6 marks. Method marks are often awarded even if your final answer is wrong.

在 Edexcel IGCSE 考试中，二次方程问题出现在 Paper 1 和 Paper 2 中，可能占 2 到 6 分。即使最终答案错误，常常也会给方法分。

Always show clear working. If you use a calculator, write down the substituted expression before giving the answer.

务必展示清晰的计算过程。如果使用计算器，先写下代入后的表达式再给出答案。

Know when to use each method:

知道何时使用哪种方法：

Method	When to use
Factorisation \| 因式分解	When the quadratic is easy to factorise, coefficient of x² is often 1 or a small number.
Quadratic formula \| 求根公式	When factorisation is difficult or impossible.
Completing the square \| 配方法	When you need the turning point or when a question asks for a solution in surd form.
Graph \| 图像	When the question gives a graph or asks you to draw one.

A calculator can check your roots quickly, but you still need to show the algebraic method for full marks.

计算器可以快速检验你的根，但你仍需展示代数方法才能得满分。

11. Practice Problems | 练习题目

Try these problems on your own after reading this article:

阅读本文后，请独立尝试这些题目：

Solve x² − 7x + 12 = 0 by factorisation.
用因式分解解 x² − 7x + 12 = 0。
Solve 3x² + 2x − 1 = 0 using the quadratic formula.
用求根公式解 3x² + 2x − 1 = 0。
Write x² − 8x + 6 in the form (x − p)² + q, and find the minimum point.
将 x² − 8x + 6 写成 (x − p)² + q 的形式，并求最小值点。
Find the discriminant of x² + 5x + 3 = 0, and state how many real roots it has.
求 x² + 5x + 3 = 0 的判别式，并说明它有几个实数根。
The sum of a number and its square is 42. Find the number.
一个数与它的平方之和为 42，求这个数。

Answers are easy to check with a calculator. If you get stuck, review the relevant section above.

答案很容易用计算器检验。如果卡住，请回顾上面相关章节。

12. Conclusion | 总结

Solving quadratic equations is a skill that connects many parts of IGCSE Mathematics. Master the four methods: factorisation, quadratic formula, completing the square, and graphing. Understand the discriminant, and always show your working in the exam.

解二次方程是一项连接 IGCSE 数学许多章节的技能。掌握四种方法：因式分解、求根公式、配方法和图像法。理解判别式，并在考试中始终展示你的解题过程。

With regular practice, you will become fast and accurate. Keep a list of mistakes you have made and review it before the test.

通过定期练习，你将变得又快又准确。把你犯过的错误列一个清单，并在考试前复习。

Good luck with your Edexcel IGCSE Mathematics exam!

祝你在 Edexcel IGCSE 数学考试中好运！

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Solving Quadratic Equations | 解二次方程